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A NOTE ON OSTROWSKI LIKE INEQUALITIES
B.G. PACHPATTE
57 Shri Niketan Colony
Near Abhinay Talkies
Aurangabad 431 001
(Maharashtra) India
EMail: bgpachpatte@hotmail.com
c 2000 Victoria University
ISSN (electronic): 1443-5756
241-04 volume 6, issue 4, article 114,
2005.
Received 13 December, 2004; accepted 23 August, 2005.
Communicated by: A. Sofo
Abstract
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Abstract
The aim of this note is to establish new Ostrowski like inequalities by using a fairly elementary analysis.
2000 Mathematics Subject Classification: 26D10, 26D15.
inequality.
1 Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2 Main Results
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
References
A Note on Ostrowski Like
Inequalities
B.G. Pachpatte
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In an elegant note [ 5 ], A.M. Ostrowski proved the following interesting and useful inequality (see also [ 3 , p. 468]):
(1.1) f ( x ) −
1 b − a
Z b a f ( t ) dt ≤
"
1
4
+ x − a + b
2
( b − a )
2
2
#
( b − a ) k f
0 k
∞
, for all x ∈ [ a, b ] , where f : [ a, b ] ⊆ entiable on ( a, b ) , whose derivative f
R
0 k f
0 k
∞
= sup x ∈ ( a,b )
| f
0
( x ) | < ∞ .
→
R is continuous on [ a, b ] and differ-
: ( a, b ) →
R is bounded on ( a, b ) , i.e.,
In the last few years, the study of such inequalities has been the focus of great attention to many researchers and a number of papers have appeared which
deal with various generalizations, extensions and variants, see [ 2 , 3 , 6 ] and the
references given therein. Inspired and motivated by the recent work going on
related to the inequality ( 1.1
), in the present note , we establish new inequalities of the type ( 1.1
) involving two functions and their derivatives. An interesting
feature of our results is that they are presented in an elementary way and provide new estimates on these types of inequalities.
A Note on Ostrowski Like
Inequalities
B.G. Pachpatte
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Our main result is given in the following theorem.
Theorem 2.1. Let entiable on ( a, b ) f, g : [ a, b ] →
R be continuous functions on [ a, b ] and differ-
, whose derivatives f
0
, g
0
: ( a, b ) →
R are bounded on ( a, b ) , i.e., k f
0 k
∞
= sup x ∈ ( a,b )
| f
0
( x ) | < ∞ , k g
0 k
∞
= sup x ∈ ( a,b )
| g
0
( x ) | < ∞ .
Then
(2.1) f ( x ) g ( x ) −
≤
1
2
2 ( b
1
− a
{| g ( x ) | k f
0 k
∞
) g ( x )
Z a b f ( y ) dy + f ( x )
+ | f ( x ) | k g
0 k
∞
}
"
1
4
+
Z b g ( y ) dy a x − a + b
2
( b − a )
2
2 #
( b − a ) , for all x ∈ [ a, b ] .
Proof. For any x, y ∈ [ a, b ] we have the following identities:
(2.2) f ( x ) − f ( y ) =
Z x f
0
( t ) dt y and
(2.3) g ( x ) − g ( y ) =
Z x g
0
( t ) dt.
y
Multiplying both sides of ( 2.2
) and ( 2.3
) by
g ( x ) and f ( x ) respectively and adding we get
(2.4) 2 f ( x ) g ( x ) − [ g ( x ) f ( y ) + f ( x ) g ( y )]
A Note on Ostrowski Like
Inequalities
B.G. Pachpatte
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= g ( x )
Z x y f
0
( t ) dt + f ( x )
Z x y g
0
( t ) dt.
Integrating both sides of ( 2.4
) with respect to
y over [ a, b ] and rewriting we have
(2.5) f ( x ) g ( x ) −
=
2 ( b
1
1
2 ( b − a )
− a )
Z b
Z b
Z a g ( x ) b f ( y ) dy + f ( x ) g ( y ) dy a g ( x )
Z x y a f
0
( t ) dt + f ( x )
Z x y g
0
( t ) dt dy.
From ( 2.5
) and using the properties of modulus we have
1 f ( x ) g ( x ) −
2 ( b − a )
≤
=
1
2 ( b − a )
1
2 ( b − a ) g ( x )
Z
{| g ( x ) | k f
0 k
∞ a b f ( y ) dy + f ( x )
Z a b g ( y ) dy
Z b
{| g ( x ) | k f
0 k
∞
| x − y | + | f ( x ) | k g
0 k
∞
| x − y |} dy a
+ | f ( x ) | k g
0 k
∞
}
"
( x − a )
2
+ ( b − x )
2
#
2
=
1
2
{| g ( x ) | k f
0 k
∞
+ | f ( x ) | k g
0 k
∞
}
"
1
4
+ x − a + b
2
( b − a )
2
2 #
( b − a ) .
The proof is complete.
Remark 1. We note that, by taking g ( x ) = 1 and hence g
0
( x ) = 0 in Theorem
2.1
, we recapture the well known Ostrowski’s inequality in ( 1.1
).
A Note on Ostrowski Like
Inequalities
B.G. Pachpatte
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Integrating both sides of ( 2.5
) with respect to
x over [ a, b ] , rewriting the resulting identity and using the properties of modulus, we obtain the following
Grüss type inequality:
(2.6)
1 b − a
Z b f ( x ) g ( x ) dx − a
1 Z b Z b
≤
2 ( b − a )
2 a a b
1
− a
Z a b f ( x ) dx b
1
− a
Z a b g ( x ) dx
{| g ( x ) | k f
0 k
∞
+ | f ( x ) | k g
0 k
∞
} | x − y | dy dx.
For other inequalities of the type ( 2.6
), see the book [ 3 ], where many other
references are given.
A slight variant of Theorem
2.1
is embodied in the following theorem.
Theorem 2.2. Let f, g, f
0
, g
0 be as in Theorem
2.1
. Then
1 Z b
(2.7) f ( x ) g ( x ) −
+ f ( x ) a g ( x ) f ( y ) dy b −
Z b a a
≤
1 b − a
1 Z b g ( y ) dy + f ( y ) g ( y ) dy b − a k f
0 k
∞ a k g
0 k
∞
"
( x − a )
3
+ ( b − x )
3
#
,
3 for all x ∈ [ a, b ] .
Proof. From the hypotheses, the identities ( 2.2
) and ( 2.3
) hold. Multiplying the
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Inequalities
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left and right sides of ( 2.2
) and ( 2.3
) we get
(2.8) f ( x ) g ( x ) − [ g ( x ) f ( y ) + f ( x ) g ( y )] + f ( y ) g ( y )
=
Z x f
0
( t ) dt
Z x g
0
( t ) dt .
y y
Integrating both sides of ( 2.8
) with respect to
y over [ a, b ] and rewriting we have
1 Z b
(2.9) f ( x ) g ( x ) −
+ f ( x ) a g ( x ) f ( y ) dy b − a
Z b a
=
1 b − a
1 Z b g ( y ) dy + a y f ( y ) g ( y ) dy b − a
Z b Z x a f
0
( t ) dt
Z x y g
0
( t ) dt dy.
From ( 2.9
) and using the properties of modulus we obtain
≤
=
1 Z b f ( x ) g ( x ) − g ( x ) f ( y ) dy b − a a
Z b
1 Z b
1
1 b − a
+ f ( x ) g ( y ) dy + f ( y ) g ( y ) dy b − a a b − a a k f
0 k
∞ k g
0 k
∞ k f
0 k
∞ k g
0 k
∞
Z b
| x − y a
"
( x − a )
3
|
2 dy
+ ( b − x )
3
#
.
3
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Inequalities
B.G. Pachpatte
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The proof is complete.
Remark 2. Integrating both sides of ( 2.9
) with respect to
x over [ a, b ] , rewriting the resulting identity, using the properties of modulus and by elementary calculations we get
(2.10)
1 b − a
Z b f ( x ) g ( x ) dx − a
1 b − a
Z b a
≤
1 Z b f ( x ) dx g ( x ) dx b − a a
1
12
( b − a )
2 k f
0 k
∞ k g
0 k
∞
.
Here, it is to be noted that the inequality ( 2.10
) is the well known ˇ
inequality (see [ 4 , p. 297]).
A Note on Ostrowski Like
Inequalities
B.G. Pachpatte
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[1] S.S. DRAGOMIR, Some integral inequalities of Grüss type, Indian J. Pure
and Appl. Math., 31 (2000), 379–415.
[2] S.S.DRAGOMIR
AND
Th.M. RASSIAS (Eds.), Ostrowski Type Inequali-
ties and Applications in Numerical Integration, Kluwer Academic Publishers, Dordrecht , 2002.
CARI ´
AND
A.M. FINK, Inequalities for
Functions and their Integrals and Derivatives, Kluwer Academic Publishers, Dordrecht, 1994.
CARI ´
AND
A.M. FINK, Classical and New
Inequalities in Analysis, Kluwer Academic Publishers, Drodrecht, 1993.
[5] A.M. OSTROWSKI, Über die Absolutabweichung einer differentiebaren
Funktion von ihrem Integralmitelwert, Comment. Math. Helv., 10 (1938),
226–227.
[6] B.G. PACHPATTE, On a new generalization of Ostrowski’s inequality, J.
Inequal. Pure and Appl. Math., 5(2) (2004), Art. 36. [ONLINE: http:
//jipam.vu.edu.au/article.php?sid=378 ]
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Inequalities
B.G. Pachpatte
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